ترغب بنشر مسار تعليمي؟ اضغط هنا

Scattering for the inhomogeneous nonlinear Schrodinger equation with a potential

126   0   0.0 ( 0 )
 نشر من قبل Sheng Wang
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

In this paper, we show the scattering of the solution for the focusing inhomogenous nonlinear Schrodinger equation with a potential begin{align*} ipartial_t u+Delta u- Vu=-|x|^{-b}|u|^{p-1}u end{align*} in the energy space $H^1(mathbb R^3)$. We prove a scattering criterion, and then we use it together with Morawetz estimate to show the scattering theory.



قيم البحث

اقرأ أيضاً

In this paper we consider the inhomogeneous nonlinear Schrodinger equation $ipartial_t u +Delta u=K(x)|u|^alpha u,, u(0)=u_0in H^s({mathbb R}^N),, s=0,,1,$ $Ngeq 1,$ $|K(x)|+|x|^s| abla^sK(x)|lesssim |x|^{-b},$ $0<b<min(2,N-2s),$ $0<alpha<{(4-2b)/(N- 2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $alpha_0(b)<alpha<(4-2b)/N$. The value $alpha_0(b)$ is the positive root of $Nalpha^2+(N-2+2b)alpha-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=mu|x|^{-b}$, $muin mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $alpha$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.
72 - Chengbin Xu 2021
In this paper, we study the long-time behavior of global solutions to the Schrodinger-Choquard equation $$ipartial_tu+Delta u=-(I_alphaast|cdot|^b|u|^{p})|cdot|^b|u|^{p-2}u.$$ Inspired by Murphy, who gave a simple proof of scattering for the non-ra dial inhomogeneous NLS, we prove scattering theory below the ground state for the intercritical case in energy space without radial assumption.
250 - Jianqing Chen , Yue Liu 2010
We study the instability of standing-wave solutions $e^{iomega t}phi_{omega}(x)$ to the inhomogeneous nonlinear Schr{o}dinger equation $$iphi_t=-trianglephi+|x|^2phi-|x|^b|phi|^{p-1}phi, qquad inmathbb{R}^N, $$ where $ b > 0 $ and $ phi_{omega} $ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $omega $ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $
We prove sharp $L^infty$ decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schrodinger equation with small initial data chosen from a weighted Sobolev space. Specifically, we work with an averaged version of the d ispersion-managed NLS in the strong dispersion management regime. The proof adapts techniques from Hayashi-Naumkin and Kato-Pusateri, which established small-data modified scattering for the standard $1d$ cubic NLS.
We consider the $1d$ cubic nonlinear Schrodinger equation with a large external potential $V$ with no bound states. We prove global regularity and quantitative bounds for small solutions under mild assumptions on $V$. In particular, we do not require any differentiability of $V$, and make spatial decay assumptions that are weaker than those found in the literature (see for example cite{Del,N,GPR}). We treat both the case of generic and non-generic potentials, with some additional symmetry assumptions in the latter case. Our approach is based on the combination of three main ingredients: the Fourier transform adapted to the Schrodinger operator, basic bounds on pseudo-differential operators that exploit the structure of the Jost function, and improved local decay and smoothing-type estimates. An interesting aspect of the proof is an approximate commutation identity for a suitable notion of a vectorfield, which allows us to simplify the previous approaches and extend the known results to a larger class of potentials. Finally, under our weak assumptions we can include the interesting physical case of a barrier potential as well as recover the result of cite{MMS} for a delta potential.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا